Generic and Non‐Generic Solutions

If you have an equation like 2x==0, it is perfectly clear that the only possible solution is x->0. However, if you have an equation like ax==0, things are not so clear. If a is not equal to zero, then x->0 is again the only solution. However, if a is in fact equal to zero, then any value of x is a solution. You can see this by using Reduce.

Solve implicitly assumes that the parameter a does not have the special value 0.

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Reduce, on the other hand, gives you all the possibilities, without assuming anything about the value of a.

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A basic difference between Reduce and Solve is that Reduce gives all the possible solutions to a set of equations, while Solve gives only the generic ones. Solutions are considered "generic" if they involve conditions only on the variables that you explicitly solve for, and not on other parameters in the equations. Reduce and Solve also differ in that Reduce always returns combinations of equations, while Solve gives results in the form of transformation rules.

Reduce gives the full version, which includes the possibility a==b==0. In reading the output, note that && has higher precedence than ||.

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Here is the full solution to a general quadratic equation. There are three alternatives. If a is nonzero, then there are two solutions for x, given by the standard quadratic formula. If a is zero, however, the equation reduces to a linear one. Finally, if a, b, and c are all zero, there is no restriction on x.

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When you have several simultaneous equations, Reduce can show you under what conditions the equations have solutions. Solve shows you whether there are any generic solutions.

This shows there can never be any solution to these equations.

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There is a solution to these equations, but only when a has the special value1.